A pendulum is swinging next to a wall. The distance from the bob of the swinging pendulum to the wall varies in a periodic way that can be modeled by a trigonometric function. The function has period $0.8$ seconds, amplitude $6 \text{ cm}$, and midline $H = 15 \text{ cm}$. At time $t = 0.5$ seconds, the bob is at its midline, moving towards the wall. Find the formula of the trigonometric function that models the distance $H$ from the pendulum's bob to the wall after $t$ seconds. Define the function using radians. $ H(t) = $
Solution: Let's start by writing a formula for the distance from the bob to the wall $u$ seconds after it passes its midline ( $u$ seconds after the time $t = 0.5$ ). Both sine and cosine can be used to model periodic contexts. We can decide which is better fitting by considering the $y$ -intercept. The sine function intercepts the $y$ -axis at its midline, and the cosine function intercepts the $y$ -axis at its peak. That way, since the bob is at its midline at time $u = 0$, we can use a sine function to model its distance, since sine functions also pass their midline at $u = 0$. The bob's distance from the wall is decreasing at time $u= 0$, while $\sin u$ is increasing at time $u= 0$, so we'll have to flip $\sin u$ vertically. Since the ordinary sine function $f(u) = \sin u$ has period $2\pi$, midline $y = 0$, and amplitude $1$, we can stretch it horizontally by a factor of ${\dfrac{0.8}{2\pi}}$, stretch it vertically by a factor of ${6}$ and flip it vertically, and move it up ${15}$ units: $ H(u) = {-6}\sin\left({\dfrac{2\pi}{0.8}}u\right) + {15}$ Since the bob passes its midline $0.5$ seconds after the stopwatch is started, $t$ seconds after the stopwatch is started is $t - 0.5$ seconds after it passes its midline, so $u = t - 0.5$. $ H(t) = {-6}\sin\left({\dfrac{2\pi}{0.8}}(t-0.5)\right) + {15}$ The function $ H(t) = {-6}\sin\left({\dfrac{2\pi}{0.8}}(t-0.5)\right) + {15}$ has period $0.8$, amplitude $6$, and midline $y = 15$, and it is decreasing past its midline at time $t= 0.5$, so it's a good model of the distance from the bob to the wall.